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Math Help - 2D Gaussian Noise

  1. #1
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    2D Gaussian Noise

    I generated a 2d Gaussian noise then modulated it with a 2d cosine (f(x,y)=cos(2*pi/N*a*x)) witch oscillate only with the x direction and "a" is the frequency.

    In a nutshell
    N: Gaussian Noise size N*N
    P: f(x,y)=cos(2*pi/N*a*x) a: frequency
    I=P.N

    after i take the Fourier transform of I, frequency pairs at offset (u,v) and (u+2a,v) have correlated phase.

    If i do the experiment with another noise for example uniform noise (in Matlab: rand(N,N)) and then modulate it with the same cosine there
    is no such correlations.

    I want to what property of Gaussian noise cause this correlations?

    Thank u all for help in advance.
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  2. #2
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    Re: 2D Gaussian Noise

    Hey peeyman.

    First question is your f(x,y) is only a function of x: what is the y? Second question is the Noise determined by P in any way (i.e. is P a function of N or N a function of P in any way)?

    If the second question is no, then the correlation should be 0 or at least if you test with a simulation, it should get close to zero is your sample size increases.
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  3. #3
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    Re: 2D Gaussian Noise

    ok sorry i made a mistake about the second question
    but first question i wrote f(x,y)=cos(2*pi/N*a*x) means f(x,y)=cos(2*pi/n*a*x+2*pi/n*0*y) witch means that frequency of y is zero so it is 2D.

    but the second question is about my mistake. I correct it as follow.
    N: Gaussian Noise size n*n
    P: f(x,y)=cos(2*pi/n*a*x) a: frequency
    I=P.N

    in the about "n" is size but N is noise witch means that P and N are not functions of each other.

    sorry for the mistake.
    I will be appreciate if have an answer for that.
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  4. #4
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    Re: 2D Gaussian Noise

    There should be no correlation between the variables if they are independent.

    Cov(X,Y) = 0 if X is independent from Y and it looks like there is a mistake if you get some kind of correlation theoretically when you derive it.

    Maybe you are talking about a different kind of correlation though: what kind of correlation are you talking about?
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  5. #5
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    Re: 2D Gaussian Noise

    i don't talk about correlation between P and N.

    I=P.N
    I take the Fourier transform of I
    FI=Fourier of I

    then i say that frequency pair at offset (u,v) and (u+2a,v) in FI are correlated.
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  6. #6
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    Re: 2D Gaussian Noise

    Can you show the calculation? Whether it's hand-written or computer generated (like in Maple/Mathematica)?
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  7. #7
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    Re: 2D Gaussian Noise

    hear is the code in matlab

    %------------------------------------------------------------
    clear all
    clc
    len=1000;
    x=[0:len];
    y=[0:len];
    [X,Y]=meshgrid(x,y);
    P=cos(4*2*pi/(len+1)*X);


    N=wgn(len+1,len+1,0);



    IF=fft2(I);
    figure
    imshow(angle(IF),[]);

    ang1=angle(IF);

    offset=8;

    for i=1:len-offset
    phase1(1,i)=ang1(1,i);
    phase1(2,i)=ang1(1,i+offset);
    end
    plot(phase1(1,,phase1(2,,'ro');
    corrcoef(phase1(1,',phase1(2,')
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  8. #8
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    Re: 2D Gaussian Noise

    %------------------------------------------------------------
    clear all
    clc
    len=1000;
    x=[0:len];
    y=[0:len];
    [X,Y]=meshgrid(x,y);
    P=cos(4*2*pi/(len+1)*X);


    N=wgn(len+1,len+1,0);

    I=P.*N


    IF=fft2(I);
    figure
    imshow(angle(IF),[]);

    ang1=angle(IF);

    offset=8;

    for i=1:len-offset
    phase1(1,i)=ang1(1,i);
    phase1(2,i)=ang1(1,i+offset);
    end
    plot(phase1(1,,phase1(2,,'ro');
    corrcoef(phase1(1,',phase1(2,')
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  9. #9
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    Re: 2D Gaussian Noise

    I'm afraid I can't really give you a suggestion for why this is the case.

    The only thing I could suggest is to look at the spectral attributes of the noise and see if there are any similarities or things that stick out with regard to your function.

    Since you are modulating the two functions, the first thing that comes to mind is to look at some kind of convolution properties of your modulated result in a way to see how the spectral attributes of the noise are related to the that of the function and look for how this would affect the correlation component you observe.
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  10. #10
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    Re: 2D Gaussian Noise

    Thank you for your time
    I really tried to find the attributes before but it is difficult to find out
    If you try it with rand not with wgn the correlation does not exist hence the
    Correlation is come somehow from the Gaussian property of the noise
    And I guess it is related to the equal energy in each frequency of the Gaussian noise
    But I dont know exactly how they are related

    However thank u for your time
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